| L(s) = 1 | − 2-s − 3-s + 4-s − 0.332·5-s + 6-s − 7-s − 8-s + 9-s + 0.332·10-s − 2.61·11-s − 12-s + 14-s + 0.332·15-s + 16-s − 3.88·17-s − 18-s − 5.61·19-s − 0.332·20-s + 21-s + 2.61·22-s + 4.21·23-s + 24-s − 4.88·25-s − 27-s − 28-s − 1.18·29-s − 0.332·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.148·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.105·10-s − 0.787·11-s − 0.288·12-s + 0.267·14-s + 0.0859·15-s + 0.250·16-s − 0.942·17-s − 0.235·18-s − 1.28·19-s − 0.0744·20-s + 0.218·21-s + 0.556·22-s + 0.878·23-s + 0.204·24-s − 0.977·25-s − 0.192·27-s − 0.188·28-s − 0.220·29-s − 0.0607·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3596409457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3596409457\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.332T + 5T^{2} \) |
| 11 | \( 1 + 2.61T + 11T^{2} \) |
| 17 | \( 1 + 3.88T + 17T^{2} \) |
| 19 | \( 1 + 5.61T + 19T^{2} \) |
| 23 | \( 1 - 4.21T + 23T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 + 0.576T + 37T^{2} \) |
| 41 | \( 1 - 0.521T + 41T^{2} \) |
| 43 | \( 1 - 3.07T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 9.71T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 0.944T + 71T^{2} \) |
| 73 | \( 1 + 4.66T + 73T^{2} \) |
| 79 | \( 1 + 0.943T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 3.38T + 89T^{2} \) |
| 97 | \( 1 - 5.81T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.035264112782653733386582399430, −7.12508540804608491296847251122, −6.71912402514702407271845664778, −5.93318714528463064605862699434, −5.24588724658016053393128643648, −4.38386508450976286640729030746, −3.53925696686604960748099609246, −2.49464021138862606037734747323, −1.73766867214685072765930940353, −0.33506363017672207921221276670,
0.33506363017672207921221276670, 1.73766867214685072765930940353, 2.49464021138862606037734747323, 3.53925696686604960748099609246, 4.38386508450976286640729030746, 5.24588724658016053393128643648, 5.93318714528463064605862699434, 6.71912402514702407271845664778, 7.12508540804608491296847251122, 8.035264112782653733386582399430
